Lie Algebras

Prerequisites: Basic
linear algebra and elementary theory of rings and fields.

We will study the theory of Lie algebras and their representations over the complex numbers. After learning the basic notions of ideals, homomorphisms, representations, etc., we will study the structure theory of nilpotent and solvable Lie algebras, and then the Cartan-Killing theory of semisimple Lie algebras, and the classification theorem in terms of root systems and Dynkin diagrams. Finally we will study the theory of finite dimensional representations of semisimple Lie algebras, including characters, the Weyl formula, cohomology, etc. The course is exclusively algebraic but connections with Lie group theory and geometry will be mentioned.

References:
HUMPHREYS, J.E. – Introduction to Lie algebras and representation theory (Springer).
KNAPP, A. W. – Lie groups beyond an introduction (Birkhäuser).
KAC, V. G. MIT class notes http://math.mit.edu/classes/18.745/classnotes.html

Note: This course is offered as a master’s degree. At the PhD, it has additional requirements.

* Standard program. The teacher has the autonomy to make any changes.